Suppose I am given a subset of $2^\omega\times\omega^\omega$ of some bounded Borel rank. Can I get an analytic uniformization of this set?
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$\begingroup$ Does analytic uniformization means that graph is an analytical set, or that the map is analytically measurable? $\endgroup$– SBFCommented Jul 18, 2013 at 20:04
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$\begingroup$ I think it means that the graph is an analytical set. $\endgroup$– jonathanvernerCommented Jul 19, 2013 at 6:39
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2$\begingroup$ A function with analytic graph is actually a Borel function. To see this, note that $f^{-1}(B)=\pi\big((X\times B)\cap\textrm{Graph} f\big)$, is analytic when $B$ is Borel. But for the same reason, $f^{-1}(B^C)$ is analytic. Since $f^{-1}(B)^C=f^{-1}(B^C)$, this shows that the preimage of a Borel set is both analytic and coanalytic, hence Borel. $\endgroup$– Michael GreineckerCommented Jul 19, 2013 at 7:46
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1 Answer
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There is an arithmetical set $A\subseteq 2^{<\omega}\times \omega^{<\omega}$ so that for any $x\in 2^{\omega}$, $A(x)=\{\sigma\mid \exists n(x|n,\sigma)\in A\}$ is an $x$-recursive tree which has an infinite path but no infinite path hyperarithmetic in $x$.
Now let $B$ be an arithmetical set so that $(x,y)\in B$ if and only if $y$ is an infinite path through $A(x)$. $B$ has the following property:
(1). For any $x$, there is some $y$ so that $(x, y)\in B$;
(2) No analytic function uniformizing $B$. Suppose not, then there is a real $z$ and $\Sigma^1_1(z)$-function $f$ uniformizing $B$. Let $x\geq_h z$, then $f(x)\leq_h x\oplus z\leq_h x$, a contradiction.
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$\begingroup$ I assume that the first formula should read $A\subseteq 2^{<\omega}\times\omega^{<\omega}$... Also, does the existence of $A$ follow from some theorem or should it be obvious (sorry, I am not very fluent in descriptive set theory let alone effective descriptive ST). $\endgroup$ Commented Aug 27, 2013 at 15:01
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$\begingroup$ You are right. I corrected the typo. The existence of $A$ follows from a well known fact that there is a $\Sigma^1_1$ set which does not contain a hyperarithmetic real. $\endgroup$– 喻 良Commented Aug 27, 2013 at 22:37